Question regarding locus of midpoint of chord of circle.

Hi everyone. The full question is as follows:

A circle passes through . Find:
(a)The equation of the circle
(b) the length of the minor arc BC
(c) the equation of the circle on AB as diameter.
(d) A Line of variable gradient , cuts the circle ABC in two points and Find in Cartesian form the equation of the locus of the midpoint of LM.

I have answered A through C, but (d) eludes me, and I can't even begin.
The answer for (a) is:

For (b):

For (c):

I know that the answer to (d) is . I also know that the line "rotates" around the point

Re: Question regarding locus of midpoint of chord of circle.

Originally Posted by LimpSpider

I know that the answer to (d) is .

I am not sure you have have y + 3 as a factor. All points (x, -3) satisfy this equation, and they are not even inside the circle.

Here is one solution to (d). Let's lower the circle by 1 so that the center is at (0, 0). Suppose the middle of a chord LM has coordinates (x0, y0). Then the line from (0, 0) to (x0, y0) is perpendicular to LM, so the line LM has the equation xx0 + yy0 = c for some constant c. (If this is unclear, please say so.) Since (x0, y0) ∈ LM, . Now, since (0, -4) ∈ line LM, . This is the required equation in x0 and y0, only the locus has to lifted back up by 1.